Page 116 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

4.1 Classical treatment 107

attached to the other end. The spring is assumed to obey Hooke's law, that is to
say, equation (4.1). The constant k is then often called the spring constant.
In classical mechanics the particle obeys Newton's second law of motion
2
d x
F ˆ ma ˆ m (4:2)
dt 2
where a is the acceleration of the particle and t is the time. The combination of
equations (4.1) and (4.2) gives the differential equation
2
d x k
ˆÿ x
dt 2 m
for which the solution is
x ˆ A sin(2ðít ‡ b) ˆ A sin(ùt ‡ b) (4:3)
where the amplitude A of the vibration and the phase b are the two constants
of integration and where the frequency í and the angular frequency ù of
vibration are related to k and m by
r 
k
ù ˆ 2ðí ˆ (4:4)
m
According to equation (4.3), the particle oscillates sinusoidally about the origin
with frequency í and maximum displacement A.
The potential energy V of a particle is related to the force F acting on it by
the expression
dV
F ˆÿ
dx
Thus, from equations (4.1) and (4.4), we see that for a harmonic oscillator the
potential energy is given by
2
2 2
1
1
V ˆ kx ˆ mù x (4:5)
2 2
The total energy E of the particle undergoing harmonic motion is given by
2 2
2
2
1
1
1
E ˆ mv ‡ V ˆ mv ‡ mù x (4:6)
2 2 2
where v is the instantaneous velocity. If the oscillator is undisturbed by outside
forces, the energy E remains ®xed at a constant value. When the particle is at
maximum displacement from the origin so that x ˆÐ A, the velocity v is zero
and the potential energy is a maximum. As jxj decreases, the potential
decreases and the velocity increases keeping E constant. As the particle crosses
p 
the origin (x ˆ 0), the velocity attains its maximum value v ˆ 2E=m.
To relate the maximum displacement A to the constant energy E, we note
that when x ˆÐ A, equation (4.6) becomes
2
1
E ˆ mù A 2
2
so that